My Math Forum The range of a matrix transformation

 Linear Algebra Linear Algebra Math Forum

 June 16th, 2010, 05:57 PM #1 Member   Joined: Feb 2009 Posts: 76 Thanks: 0 The range of a matrix transformation How to do you determine if a vector (w) is in the range a matrix transformation. Let f: R^2 --> R^3 f(x) = Ax $\ A=\begin{bmatrix} 1 &2 \\ 0 &1 \\ 1& 1 \end{bmatrix}$/extract_itex] Vector w $\ w=\begin{bmatrix} 1\\ 1\\ 1 \end{bmatrix}\$ Why is it a no? My work: ( I don't know if I'm doing it right) $\ f(x)=\begin{bmatrix} 1 &2 \\ 0 &1 \\ 1 & 1 \end{bmatrix}\begin{bmatrix} x\\ y \end{bmatrix}\$ $\ =\begin{bmatrix} 1 &2 \\ 0& 1 \end{bmatrix}\begin{bmatrix} 1\\ 1 \end{bmatrix}=\begin{bmatrix} 3\\ 1 \end{bmatrix}\$ and why is for Vector w, it's a yes? $\ w=\begin{bmatrix} 8\\ 5\\ 3 \end{bmatrix}\$ $\ =\begin{bmatrix} 1 &2 \\ 0& 1 \end{bmatrix}\begin{bmatrix} 8\\ 5 \end{bmatrix}=\begin{bmatrix} 18\\ 5 \end{bmatrix}\$ The book says that the set of all images of the vectors in R^n is called the range of f. I don't know what it means. Can someone please explain to me how to determine the range. Thanks in advance  June 16th, 2010, 07:21 PM #2 Senior Member Joined: Oct 2007 From: Chicago Posts: 1,701 Thanks: 3 Re: The range of a matrix transformation You want w= f(v) = Av, where v is a vector $\[\begin{array}x\\ y\end{array}$$. The equation should be: $$\begin{array}1\\ 1\\ 1\end{array}$ = $\begin{array} 1&2\\ 0&1\\ 1&1\end{array}$ $\begin{array}x\\ y\end{array}$$ You cannot substitute values of w into v, you need to see if any value of v, when multiplied by A gives you w. See if there is any v=[x,y] which satisfies this equation. Edit: Some clarification. "The set of all images" is the set $\{f(v)|v\in \mathbb{R}^2\}$. So, if w is in the image, w=f(v) for some vector v in R^2. To check if such a vector exists, see if there is any vector which satisfies f(v)=w.

 Tags matrix, range, transformation

,
,
,

,

,

,

,

,

,

,

,

,

,

,

# how to check if a vector is in range ofa transformation

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post tomtong Algebra 0 June 10th, 2013 05:53 PM Niko Bellic Linear Algebra 5 January 3rd, 2013 11:32 AM aliya Linear Algebra 1 August 10th, 2010 12:39 PM remeday86 Linear Algebra 3 July 25th, 2010 04:44 PM sansar Linear Algebra 1 April 7th, 2009 10:07 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top